A115135 - OEIS

%I #15 Sep 08 2022 08:45:23

%S 0,108,1407,1851,2407,9768,12340,15568,58435,73423,92235,342076,

%T 429432,539076,1995255,2504403,3143455,11630688,14598220,18322888,

%U 67790107,85086151,106795107,395111188,495919920,622448988,2302878255,2890434603

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+617, y).

%C Corresponding values y of solutions (x, y) are in A160176.

%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 0.

%H G. C. Greubel, <a href="/A115135/b115135.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).

%F a(n) = 6*a(n-3) -a(n-6) +1234 for n > 6; a(1)=0, a(2)=108, a(3)=1407, a(4)=1851, a(5)=2407, a(6)=9768.

%F G.f.: x*(108 +1299*x +444*x^2 -92*x^3 -433*x^4 -92*x^5)/((1-x)*(1 -6*x^3 +x^6)).

%F a(3*k+1) = 617*A001652(k) for k >= 0.

%t LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,108,1407,1851,2407,9768,12340}, 50] (* _G. C. Greubel_, May 04 2018 *)

%o (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689), print1(n, ",")))}

%o (PARI) x='x+O('x^30); Vec(x*(108 +1299*x +444*x^2 -92*x^3 -433*x^4 -92*x^5)/((1-x)*(1 -6*x^3 +x^6))) \\ _G. C. Greubel_, May 04 2018

%o (Magma) I:=[0,108,1407,1851,2407,9768,12340]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +self(n-7): n in [1..30]]; // _G. C. Greubel_, May 04 2018

%Y Cf. A160176, A001652, A111258, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).

%K nonn,easy

%O 1,2

%A _Mohamed Bouhamida_, Jun 03 2007

%E Edited and two terms added by _Klaus Brockhaus_, May 18 2009